v2009.01.01 - Convex Optimization

122 CHAPTER 2. CONVEX GEOMETRY
2.9.2.6.3 Theorem. Convex subsets of positive semidefinite cone.
The subsets of the positive semidefinite cone S M + , for 0≤ρ≤M
S M +(ρ) ∆ = {X ∈ S M + | rankX ≥ ρ} (232)
are pointed convex cones, but not closed unless ρ = 0 ; id est, S M +(0)= S M + .
⋄
Proof. Given ρ , a subset S M +(ρ) is convex if and only if
convex combination of any two members has rank at least ρ . That is
confirmed applying identity (229) from Lemma 2.9.2.6.1 to (1370); id est,
for A,B ∈ S M +(ρ) on the closed interval µ∈[0, 1]
rank(µA + (1 −µ)B) ≥ min{rankA, rankB} (233)
It can similarly be shown, almost identically to proof of the lemma, any conic
combination of A,B in subset S M +(ρ) remains a member; id est, ∀ζ, ξ ≥0
rank(ζA + ξB) ≥ min{rank(ζA), rank(ξB)} (234)
Therefore, S M +(ρ) is a convex cone.
Another proof of convexity can be made by projection arguments:
2.9.2.7 Projection on S M +(ρ)
Because these cones S M +(ρ) indexed by ρ (232) are convex, projection on
them is straightforward. Given a symmetric matrix H having diagonalization
H = ∆ QΛQ T ∈ S M (A.5.2) with eigenvalues Λ arranged in nonincreasing
order, then its Euclidean projection (minimum-distance projection) on S M +(ρ)
P S M
+ (ρ)H = QΥ ⋆ Q T (235)
corresponds to a map of its eigenvalues:
Υ ⋆ ii =
{ max {ǫ , Λii } , i=1... ρ
max {0, Λ ii } , i=ρ+1... M
(236)
where ǫ is positive but arbitrarily close to 0.